Alkalinity Calculation Methods

Several methods are used by the Alkalinity Calculator
to determine the carbonate and bicarbonate endpoints of your titration data.
After these equivalence points are found, the Alkalinity Calculator then
calculates the sample's alkalinity and its concentrations of hydroxide,
carbonate, and bicarbonate.

To see how the program has evolved over time, please read the
version history.

The inflection point method, also known as the incremental
equivalence method, determines the carbonate and bicarbonate endpoints
in the titration by finding the greatest change in the measured pH per
unit volume of acid added. This depends on the presence of a discernable
inflection point in the measured titration curve. If the alkalinity is
very low, such inflection points may or may not be visible in the data.

If more than one pair of data points produces the same
maximum change in pH per unit of acid added in the region of the expected
endpoint, the endpoint is set at the average titrant
volume of the first and last such points, and the endpoint pH at that
titrant volume is interpolated. This method may be different from some
other programs that you may have used in the past, but it is consistent
with the latest recommendations in the
USGS
National Field Manual.

The carbonate endpoint will not be determined if the
highest pH in the data set is less than or equal to 8.3. The search
for the carbonate endpoint is restricted to the range of measured
pH values less than pK2 and greater than pK1.
pK1 is equal to -log10(K1), where
K1 is the first acid dissociation constant of carbonic acid.
pK2 is equal to -log10(K2), where
K2 is the second acid dissociation constant of carbonic acid.
This restricts the search to the approximate pH range between 10.3 and
6.3.

The search for the bicarbonate endpoint is restricted
to the range of measured pH values less than or equal to pK1,
which is about 6.3. If no titration data are available in this pH
range, the analysis will be abandoned.

The fixed endpoint method determines the alkalinity
and other results of the titration based on one or more user-specified
endpoints. If only one endpoint is specified, the other endpoint (if
necessary) is found using the inflection point method. If necessary,
a linear interpolation will be used to determine the volume of titrant
used at the specified endpoint pH.

If a carbonate endpoint is specified, it will be used only
if the specified endpoint is within the range of the measured pH data.

The fixed endpoint method can be useful in the event that
one or more of the other methods fail to detect the correct endpoint.
Due to the fact that the location of the equivalence points cannot be
predicted accurately in advance, however, it is inadvisable to use the
fixed endpoint method to the exclusion of all other methods.

Using the theoretical equation for a carbonate titration,
this method finds the carbonate and bicarbonate endpoints separately
using a nonlinear least-squares fitting technique.

The carbonate endpoint is determined only if the highest pH
in the data set is greater than 8.3 and at least three data points
are within one pH unit of the expected endpoint. The expected endpoint is
estimated with one of several methods; the default endpoint is the average of
pK1 and pK2. Generally, this range is from about
pH 7.3 to 9.3, but decreases when the concentration of carbonate and
bicarbonate is low.

The bicarbonate endpoint will be determined only if at least
three data points are within about one pH unit from the expected endpoint.
The expected endpoint is estimated with one of several methods; the default
range is between (pK1 - 0.6) and (pK1 - 3.1) pH units.
Generally, this range is from about pH 5.8 to 3.3, but ranges higher when
the concentration of bicarbonate is low.

This method finds the best fit of the theoretical
carbonic acid titration curve near each endpoint by minimizing the
sum of squared residuals between the measured and predicted volumes of
titrant at each measured pH. Powell's conjugate direction set method,
discarding the direction of largest decrease, is used to minimize this
nonlinear function. Powell's method, as used in this program, was adapted
to Perl from the Fortran code given in Numerical Recipes by Press
et al. (1989).

The equation used to describe the theoretical carbonic acid
titration curve accounts for temperature and activity effects as well as
dilution of the sample by the titrant. This equation is:

where

Vt

is the volume of acid titrant added

Vo

is the initial volume of the sample

Ca

is the normality of the acid titrant

Kw'

is the acid dissociation constant for water

{H+}

is the hydrogen ion activity

is the activity coefficient for H+

Alko

is the alkalinity of the sample

K1'

is the first acid dissociation constant for H2CO3

K2'

is the second acid dissociation constant for H2CO3

Co

is the sum of the carbonic acid, bicarbonate, and carbonate
concentrations

The only unknown variables in the above equation are
Alko and Co. Powell's method is used to
find the best-fit values for these variables. Once these values are known,
the theoretical endpoints and the carbonate and bicarbonate concentrations
can be calculated directly.

Both temperature and activity corrections are applied to
the acid dissociation constants for water and carbonic acid. The mixed
constants used by this method are defined as:

where concentrations are indicated by square brackets
[ ], activities are denoted by braces { }, and the symbols are the activity coefficients for
the species subscripted. For more information on the chemistry of carbonic
acid in water, read Aquatic Chemistry Concepts by Pankow (1991).

Activity corrections are based on the extended
Debye-Hückel relation between activity coefficients and ionic strength.
Ionic strength (I) is estimated from specific conductance (SC) as:

I = 0.000025 * 0.59 * SC

This equation is based on a relation between ionic strength
and total dissolved solids, taken from Sawyer and McCarty (1967), which
accounts for the first constant (0.000025), and an empirical relation between
total dissolved solids and specific conductance, taken from Hem (1992),
which accounts for the second constant (0.59).

The relations used to calculate the effects of temperature
on the activity coefficients were taken from equations used in the
water-quality model CE-QUAL-W2, version 3.1 (Cole and Wells, 2002).
The acid dissociation constants are calculated as a function of temperature
according to equations given by Stumm & Morgan's Aquatic Chemistry,
3rd edition (1996):

where the coefficients of the equation are:

Coefficient

Kw

K1

K2

a1

-283.9710

-356.3094

-107.8871

a2

-0.05069842

-0.06091964

-0.03252849

a3

13323.00

21834.37

5151.79

a4

102.24447

126.8339

38.92561

a5

-1119669

-1684915

-563713.9

and T is the absolute water temperature in
degrees Kelvin. Temperature in Kelvin is obtained by adding 273.15 to
the temperature in degrees Celsius.

This method is the same as the theoretical carbonate
titration curve method 1, except that the fitting technique uses data
from the entire titration curve rather than just from the regions near
the expected endpoints.

The carbonate and bicarbonate endpoints determined by
this method are guaranteed to be consistent with one another and with the
chemistry of carbonic acid. By contrast, when the endpoints are fitted
independently, as in the best fit method 1 above, the calculated values may
not be consistent with each other. After all, method 1 above will determine
two sets of values for Alko and Co
if two separate endpoints are determined. This method avoids such an
inconsistency.

If the sample should have a significant amount of
non-carbonate alkalinity, however, the fitted titration curve may be poor.
In other words, the chemistry of carbonic acid alone may not be able to
account for the shape of the measured titration curve. Indications of a
poor fit include systematic (non-random) deviations above or below the
measured data, clear errors in the predicted endpoints, or a starting
point far from the measured starting point of the titration. The amount
of non-carbonate alkalinity can be estimated, but such estimates are
susceptible to large errors; therefore, an estimate of the non-carbonate
alkalinity is not made by this program.

This method is included specifically to give the user an
indication of whether the chemistry of carbonic acid alone can account
for the shape of the entire titration curve. If the fit is poor, it is
unlikely that the calculated endpoints will be of much use. A poor fit
does indicate, however, that the carbonate and bicarbonate concentrations
calculated by the other methods actually are accounting for more than just
carbonate and bicarbonate. An excellent fit with this method is evidence
that the titration curve is dominated by the carbonic acid species;
the calculated carbonate and bicarbonate concentrations will be fairly
accurate in that case.

Gran titration analysis can be used to determine the location
of the carbonate and bicarbonate endpoints (as well as other parameters
such as K1', K2',
and Ca) using data that are somewhat removed
from the endpoints of interest. Indeed, the linearizing assumptions used
by Gran's method are valid only for data that are some distance away
from the endpoints (Gran, 1950 and 1952). Gran's method does not rely
on the presence of inflection points in the titration curve; therefore,
it is particularly useful for waters with low alkalinity.

Gran's method also can be used to determine the "hydroxide"
endpoint, which often does not produce a visible inflection point in the
titration curve. This is the only method used by the Alkalinity Calculator
that attempts to find the hydroxide endpoint, although all of these methods
can compensate for the titration of hydroxide when calculating the
concentrations of carbonate and bicarbonate (if the
advanced speciation method is used). The titration
of hydroxide is only important in water samples with very high pH, say
greater than pH 9.2.

Gran's method can be used to quantify negative values
of alkalinity in solutions with an initial pH lower than the bicarbonate
equivalence point. It is the only method available in the Alkalinity
Calculator that can do so.

In order for Gran's method to work well, it is best to carry
out acidimetric titrations to pH values well below the final endpoint.
In other words, it works best for environmental samples if you provide
data down to pH 3.5 or lower. In addition, because Gran's method uses
data that are somewhat removed from the endpoints, it is best not to skip
over any part of the titration.

Using the known chemistry of carbonic acid and some
simplifying assumptions, Gran's method linearizes several functions that
describe parts of the titration curve. For a derivation, read Aquatic
Chemistry by Stumm and Morgan (1981) or go back to the original papers
by Gunnar Gran (1950, 1952). The Gran functions are as follows:

where

Vs

is the titrant volume at the bicarbonate endpoint

Vw

is the titrant volume at the carbonate endpoint

Vx

is the titrant volume at the hydroxide endpoint

and the other variables are defined as above. For these
functions, it is useful to know that:

Note that F2 and F3, as
well as F4 and F5, are simple algebraic
rearrangements of each other. Note also that functions F5
and F6 are rarely used, as the pH of the sample would
have to be rather high to invoke these functions.

The Gran plot is obtained by plotting each of the Gran
functions against titrant volume and fitting a line through each function's
data points in a particular pH region.

Function F1 is valid for the pH range
just beyond the bicarbonate endpoint. Functions F2
and F3 are valid somewhere in the pH range between the
carbonate and bicarbonate endpoints. Functions F4 and
F5 are valid for pH values between the carbonate and
hydroxide endpoints. Finally, function F6 is valid for
pH values higher than the hydroxide endpoint.

This program does its best to fit a line through the points
in each of these functions' linear ranges. This is not a trivial task.
If the fitted lines don't work well for your particular data set, you may
want to do your own Gran titration analysis. Before doing that, however,
please send me the data set so that
I can improve this program. The potential error in the results increases
when the lines derived from the Gran functions are based on only a few
points. Beware of results that are derived from only two points!

Once the fitted lines are obtained, the various endpoints
can be found by determining the titrant volume where each function crosses
the Y axis.

You may note that in order to calculate F2,
one needs to have a value for Vs, which is obtained
from the solution of F1. Similarly, the value of
Vw from an analysis of F2 is necessary
to plot and analyze F3. Function F4
requires both Vs and Vw, from
the results of F2 and either F1
or F3. Function F5 requires a value
for Vw from the results of either F2
or F4.

If F1 cannot be solved, then this method
estimates F3 using a zero value for Vw.
The value for Vs obtained from F3,
then, is used in calculating F2. If a value of
Vw is obtained from F2, then
F3 (and then F2) is re-evaluated.
Function F4 is only calculated if the results
from F2 and from either F1 or
F3 are available. Functions F5 and
F6 are attempted only if a value for Vw
could be found from F2. Complicated? Well, yes. But,
it seems to work most of the time.

The concentrations of carbonate and bicarbonate in the
sample are calculated using either a simple mass-balance
speciation method (deprecated-- no longer used) or an
advanced speciation method. Either way, the
calculation of the sample alkalinity is a straightforward accounting of
the acid used to neutralize the sample, and the hydroxide concentration is
easily calculated from the sample's initial pH. The equation for calculating
alkalinity is documented in section 6.6.5.A of the
USGS
National Field Manual.

The simple mass-balance method for determining concentrations
of carbonate and bicarbonate in a titrated water sample is an old approach
that is no longer used by USGS or the Alkalinity Calculator (starting with
version 2.22 // 22-Sep-2012); see USGS Office of Water Quality
Technical Memorandum
2012.05 for more information. For purposes of documentation only, the
method is described here. The simple mass-balance speciation method works
relatively well only for samples with a pH less than 9.2.
In such cases, most of the alkalinity can be attributed to bicarbonate.
The higher the sample pH, the greater the potential error.

This method assumes that all of the acid added above the
carbonate equivalence point neutralizes only carbonate, and that all of the
acid added between the carbonate and bicarbonate equivalence points
neutralizes only bicarbonate. Hydroxide is not considered. The concentration
of hydroxide may be calculated independently from the sample pH, but should
not be considered, as any alkalinity due to hydroxide is lumped by this
method into the estimated concentration of carbonate. The chemistry of
carbonic acid is ignored completely.

Despite these limitations and assumptions, this simple
method works relatively well for samples with a pH less than or
equal to 9.2. Testing with a theoretical titration indicates that if
this method is used for samples with pH less than 9.2, errors due solely
to the speciation method should be restricted to less than 10 percent, or
less than 1 mg/L.

The speciation equations used by the simple mass-balance
speciation method are:

Both A and B must be in milliliters. If the
titration is performed with a Hach digital titrator, digital counts may be
converted to milliliters by dividing the number of counts by 800.

If the value of B is determined to be less than zero
using Gran's method, the bicarbonate concentration is set to zero. If the
initial sample pH is greater than 8.3 and A is not determined, then
this method fails and the concentrations of carbonate and bicarbonate are
not calculated. Again, this method is deprecated; see USGS Office of Water
Quality Technical
Memorandum 2012.05 for more information and use the
advanced speciation method instead.

In this method, the concentrations of hydroxide, carbonate,
and bicarbonate are calculated from the sample pH and alkalinity, according
to well-known theoretical relations. Underlying these relations is the
assumption that the chemistry of carbonic acid accounts for all of the
significant alkalinity in the sample. If any other species with acid/base
properties (such as borate, silicate, ammonia, etc.) are present in
significant concentrations, then the calculated concentrations of carbonate
and bicarbonate will be in error.

The equations for hydroxide, carbonate, and bicarbonate
are as follows:

where the symbols are used as defined above,
Alko is in units of equivalents per liter, and the
concentrations of hydroxide, carbonate, and bicarbonate are in moles per
liter. To convert to milliequivalents per liter (meq/L), multiply moles
per liter by 1000 for hydroxide or bicarbonate, and by 2000 for carbonate.
To convert moles per liter to milligrams per liter (mg/L), multiply the
bicarbonate result by 61,017.1, the carbonate result by 60,009.2, and the
hydroxide result by 17,007.3. The concentrations of hydroxide, carbonate,
and bicarbonate are constrained to be non-negative.

Note that these equations are different from those that
the USGS had traditionally used for the calculation of carbonate and
bicarbonate concentrations. Those older equations (see the
simple mass-balance method above) did not account for
the possible titration of significant amounts of hydroxide, and they did
not take into account the equilibrium chemistry of the carbonate species.
These oversights made the calculations simpler, and the old equations
worked well for many samples, but those equations should not be used
for any sample with a pH greater than 9.2. The equations used in this
advanced speciation method, however, are valid for any sample pH, include
the effects of hydroxide, and consider the equilibrium chemistry of the
carbonate species. For these reasons, the advanced speciation method is the
only speciation method offered in the Alkalinity Calculator since version
2.22 // 22-Sep-2012.

U.S. Geological Survey, 2012, Replacement of the simple
speciation method for computation of carbonate and bicarbonate concentrations
from alkalinity titrations: Office of Water Quality Technical Memorandum
2012.05, accessed September 16, 2012, at
http://water.usgs.gov/admin/memo/QW/qw12.05.pdf.